On the Fourier dimension of (d,k)-sets and Kakeya sets with restricted directions
Abstract
A (d, k)-set is a subset of ℝd containing a k-dimensional unit ball of all possible orientations. Using an approach of D. Oberlin we prove various Fourier dimension estimates for compact (d, k)-sets. Our main interest is in restricted (d, k)-sets, where the set only contains unit balls with a restricted set of possible orientations Γ. In this setting our estimates depend on the Hausdorff dimension of Γ and can sometimes be improved if additional geometric properties of Γ are assumed. We are led to consider cones and prove that the cone in ℝd+1 has Fourier dimension d−1, which may be of interest in its own right.
Citation
Fraser , J , Harris , T L J & Kroon , N G 2022 , ' On the Fourier dimension of (d,k)-sets and Kakeya sets with restricted directions ' , Mathematische Zeitschrift , vol. First Online . https://doi.org/10.1007/s00209-022-02971-3
Publication
Mathematische Zeitschrift
Status
Peer reviewed
ISSN
0025-5874Type
Journal article
Description
Funding: JMF was financially supported by an EPSRC Standard Grant (EP/R015104/1) and a Leverhulme Trust Research Project Grant (RPG-2019-034).Collections
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